Skorokhod's representation theorem

In mathematics and statistics, Skorokhod's representation theorem is a result that shows that a weakly convergent sequence of probability measures whose limit measure is sufficiently well-behaved can be represented as the distribution/law of a pointwise convergent sequence of random variables defined on a common probability space. It is named for the Ukrainian mathematician A. V. Skorokhod.

Statement
Let $$(\mu_n)_{n \in \mathbb{N}}$$ be a sequence of probability measures on a metric space $$S$$ such that $$\mu_n$$ converges weakly to some probability measure $$\mu_\infty$$ on $$S$$ as $$n \to \infty$$. Suppose also that the support of $$\mu_\infty$$ is separable. Then there exist $$S$$-valued random variables $$X_n$$ defined on a common probability space $$(\Omega,\mathcal{F},\mathbf{P})$$ such that the law of $$X_n$$ is $$\mu_n$$ for all $$n$$ (including $$n=\infty$$) and such that $$(X_n)_{n \in \mathbb{N}}$$ converges to $$X_\infty$$, $$\mathbf{P}$$-almost surely.